This short note summarizes the works done in collaboration between S.
Belliard (CEA, Saclay), L. Frappat (LAPTh, Annecy), S. Pakuliak (JINR, Dubna),
E. Ragoucy (LAPTh, Annecy), N. Slavnov (Steklov Math. Inst., Moscow) and more
recently A. Hutsalyuk (Wuppertal / Moscow) and A. Liashyk (Kiev / Moscow).
It presents the construction of Bethe vectors, their scalar products and the
form factors of local operator for integrable models based on the
(super)algebras $gl_n$, $gl_{m|p}$ or their quantum deformations.
It corresponds to two talks given by E.R. and N.S. at \textsl{Correlation
functions of quantum integrable systems and beyond}, in honor of Jean-Michel
Maillet for his 60's (ENS Lyon, October 2017).

Tensor product state (TPS) based methods are powerful tools to efficiently
simulate quantum many-body systems in and out of equilibrium. In particular,
the one-dimensional matrix-product (MPS) formalism is by now an established
tool in condensed matter theory and quantum chemistry. In these lecture notes,
we combine a compact review of basic TPS concepts with the introduction of a
versatile tensor library for Python (TeNPy) [https://github.com/tenpy/tenpy].
As concrete examples, we consider the MPS based time-evolving block decimation
and the density matrix renormalization group algorithm. Moreover, we provide a
practical guide on how to implement abelian symmetries (e.g., a particle number
conservation) to accelerate tensor operations.

Free fermion systems enjoy a privileged place in physics. With their simple
structure they can explain a variety of effects, ranging from insulating and
metallic behaviours to superconductivity and the integer quantum Hall effect.
Interactions, e.g. in the form of Coulomb repulsion, can dramatically alter
this picture by giving rise to emerging physics that may not resemble free
fermions. Examples of such phenomena include high-temperature
superconductivity, fractional quantum Hall effect, Kondo effect and quantum
spin liquids. The non-perturbative behaviour of such systems remains a major
obstacle to their theoretical understanding that could unlock further
technological applications. Here, we present a pedagogical review of
"interaction distance" [Nat. Commun. 8, 14926 (2017)] -- a systematic method
that quantifies the effect interactions can have on the energy spectrum and on
the quantum correlations of generic many-body systems. In particular, the
interaction distance is a diagnostic tool that identifies the emergent physics
of interacting systems. We illustrate this method on the simple example of a
one-dimensional Fermi-Hubbard dimer.

I gently introduce the diagrammatic birdtrack notation, first for vector
algebra and then for permutations. After moving on to general tensors I review
some recent results on Hermitian Young operators, gluon projectors, and
multiplet bases for SU(N) colour space.

These notes are intended as a detailed discussion on how to implement the
diagrammatic Monte Carlo method for a physical system which is technically
simple and where it works extremely well, namely the Fröhlich polaron
problem. Sampling schemes for the Green function as well as the self-energy in
the bare and skeleton (bold) expansion are disclosed in full detail. We discuss
the Monte Carlo updates, possible implementations in terms of common data
structures, as well as techniques on how to perform the Fourier transforms for
functions with discontinuities. Control over the variety of parameters,
especially in the bold scheme, is demonstrated. Sample codes are made available
online along with extensive documentation. Towards the end, we discuss various
extensions of the method and their applications. After working through these
notes, the reader will be well equipped to explore the richness of the
diagrammatic Monte Carlo method for quantum many-body systems.

3 citations

We provide a brief but self-contained review of conformal field theory on the
Riemann sphere. We first introduce general axioms such as local conformal
invariance, and derive Ward identities and BPZ equations. We then define
minimal models and Liouville theory by specific axioms on their spectrums and
degenerate fields. We solve these theories by computing three- and four-point
functions, and discuss their existence and uniqueness.